Learning-based Optimization for Signal and Image Processing

Incorporating machine learning techniques into optimization problems and solvers attracts increasing attention. Given a particular type of optimization problem that needs to be solved repeatedly, machine learning techniques can find some features for this category of optimization and develop algorithms with excellent performance. This thesis deals with algorithms and convergence analysis in learning-based optimization in three aspects: learning dictionaries, learning optimization solvers and learning regularizers.Learning dictionaries for sparse coding is significant for signal processing. Convolutional sparse coding is a form of sparse coding with a structured, translation invariant dictionary. Most convolutional dictionary learning algorithms to date operate in the batch mode, requiring simultaneous access to all training images during the learning process, which results in very high memory usage, and severely limits the training data size that can be used. I proposed two online convolutional dictionary learning algorithms that offered far better scaling of memory and computational cost than batch methods and provided a rigorous theoretical analysis of these methods.Learning fast solvers for optimization is a rising research topic. In recent years, unfolding iterative algorithms as neural networks has become an empirical success in solving sparse recovery problems. However, its theoretical understanding is still immature, which prevents us from fully utilizing the power of neural networks. I studied unfolded ISTA (Iterative Shrinkage Thresholding Algorithm) for sparse signal recovery and established its convergence. Based on the properties of parameters required by convergence, the model can be significantly simplified and, consequently, has much less training cost and better recovery performance.Learning regularizers or priors improves the performance of optimization solvers, especially for signal and image processing tasks. Plug-and-play (PnP) is a non-convex framework that integrates modern priors, such as BM3D or deep learning-based denoisers, into ADMM or other proximal algorithms. Although PnP has been recently studied extensively with great empirical success, theoretical analysis addressing even the most basic question of convergence has been insufficient. In this thesis, the theoretical convergence of PnP-FBS and PnP-ADMM was established, without using diminishing stepsizes, under a certain Lipschitz condition on the denoisers. Furthermore, real spectral normalization was proposed for training deep learning-based denoisers to satisfy the proposed Lipschitz condition

International Conference on Continuous Optimization (ICCOPT) 2019 Conference Book

The Sixth International Conference on Continuous Optimization took place on the campus of the Technical University of Berlin, August 3-8, 2019. The ICCOPT is a flagship conference of the Mathematical Optimization Society (MOS), organized every three years. ICCOPT 2019 was hosted by the Weierstrass Institute for Applied Analysis and Stochastics (WIAS) Berlin. It included a Summer School and a Conference with a series of plenary and semi-plenary talks, organized and contributed sessions, and poster sessions. This book comprises the full conference program. It contains, in particular, the scientific program in survey style as well as with all details, and information on the social program, the venue, special meetings, and more

Inverse Problems and Self-similarity in Imaging

This thesis examines the concept of image self-similarity and provides solutions to various associated inverse problems such as resolution enhancement and missing fractal codes. In general, many real-world inverse problems are ill-posed, mainly because of the lack of existence of a unique solution. The procedure of providing acceptable unique solutions to such problems is known as regularization. The concept of image prior, which has been of crucial importance in image modelling and processing, has also been important in solving inverse problems since it algebraically translates to the regularization procedure. Indeed, much recent progress in imaging has been due to advances in the formulation and practice of regularization. This, coupled with progress in optimization and numerical analysis, has yielded much improvement in computational methods of solving inverse imaging problems. Historically, the idea of self-similarity was important in the development of fractal image coding. Here we show that the self-similarity properties of natural images may be used to construct image priors for the purpose of addressing certain inverse problems. Indeed, new trends in the area of non-local image processing have provided a rejuvenated appreciation of image self-similarity and opportunities to explore novel self-similarity-based priors. We first revisit the concept of fractal-based methods and address some open theoretical problems in the area. This includes formulating a necessary and sufficient condition for the contractivity of the block fractal transform operator. We shall also provide some more generalized formulations of fractal-based self-similarity constraints of an image. These formulations can be developed algebraically and also in terms of the set-based method of Projection Onto Convex Sets (POCS). We then revisit the traditional inverse problems of single frame image zooming and multi-frame resolution enhancement, also known as super-resolution. Some ideas will be borrowed from newly developed non-local denoising algorithms in order to formulate self-similarity priors. Understanding the role of scale and choice of examples/samples is also important in these proposed models. For this purpose, we perform an extensive series of numerical experiments and analyze the results. These ideas naturally lead to the method of self-examples, which relies on the regularity properties of natural images at different scales, as a means of solving the single-frame image zooming problem. Furthermore, we propose and investigate a multi-frame super-resolution counterpart which does not require explicit motion estimation among video sequences

Long-range interacting quantum systems

In this review recent investigations are summarized of many-body quantum systems with long-range interactions, which are currently realized in Rydberg atom arrays, dipolar systems, trapped-ion setups, and cold atoms in cavities. In these experimental platforms parameters can be easily changed, and control of the range of the interaction has been achieved. The main aim of the review is to present and identify the common and mostly universal features induced by long-range interactions in the behavior of quantum many-body systems. Discussed are the case of strong nonlocal couplings, i.e., the nonadditive regime, and the one in which energy is extensive, but low-energy, long-wavelength properties are altered with respect to the short-range case. When possible, comparisons with the corresponding results for classical systems are presented. Finally, cases of competition with local effects are also reviewed

Nonlinear Data: Theory and Algorithms

Techniques and concepts from differential geometry are used in many parts of applied mathematics today. However, there is no joint community for users of such techniques. The workshop on Nonlinear Data assembled researchers from fields like numerical linear algebra, partial differential equations, and data analysis to explore differential geometry techniques, share knowledge, and learn about new ideas and applications

Learning to compress and search visual data in large-scale systems

The problem of high-dimensional and large-scale representation of visual data is addressed from an unsupervised learning perspective. The emphasis is put on discrete representations, where the description length can be measured in bits and hence the model capacity can be controlled. The algorithmic infrastructure is developed based on the synthesis and analysis prior models whose rate-distortion properties, as well as capacity vs. sample complexity trade-offs are carefully optimized. These models are then extended to multi-layers, namely the RRQ and the ML-STC frameworks, where the latter is further evolved as a powerful deep neural network architecture with fast and sample-efficient training and discrete representations. For the developed algorithms, three important applications are developed. First, the problem of large-scale similarity search in retrieval systems is addressed, where a double-stage solution is proposed leading to faster query times and shorter database storage. Second, the problem of learned image compression is targeted, where the proposed models can capture more redundancies from the training images than the conventional compression codecs. Finally, the proposed algorithms are used to solve ill-posed inverse problems. In particular, the problems of image denoising and compressive sensing are addressed with promising results.Comment: PhD thesis dissertatio

A Study of the Structural Similarity Image Quality Measure with Applications to Image Processing

Since its introduction in 2004, the Structural Similarity (SSIM) index has gained widespread popularity as an image quality assessment measure. SSIM is currently recognized to be one of the most powerful methods of assessing the visual closeness of images. That being said, the Mean Squared Error (MSE), which performs very poorly from a perceptual point of view, still remains the most common optimization criterion in image processing applications because of its relative simplicity along with a number of other properties that are deemed important. In this thesis, some necessary tools to assist in the design of SSIM-optimal algorithms are developed. This work combines theoretical developments with experimental research and practical algorithms. The description of the mathematical properties of the SSIM index represents the principal theoretical achievement in this thesis. Indeed, it is demonstrated how the SSIM index can be transformed into a distance metric. Local convexity, quasi-convexity, symmetries and invariance properties are also proved. The study of the SSIM index is also generalized to a family of metrics called normalized (or M-relative) metrics. Various analytical techniques for different kinds of SSIM-based optimization are then devised. For example, the best approximation according to the SSIM is described for orthogonal and redundant basis sets. SSIM-geodesic paths with arclength parameterization are also traced between images. Finally, formulas for SSIM-optimal point estimators are obtained. On the experimental side of the research, the structural self-similarity of images is studied. This leads to the confirmation of the hypothesis that the main source of self-similarity of images lies in their regions of low variance. On the practical side, an implementation of local statistical tests on the image residual is proposed for the assessment of denoised images. Also, heuristic estimations of the SSIM index and the MSE are developed. The research performed in this thesis should lead to the development of state-of-the-art image denoising algorithms. A better comprehension of the mathematical properties of the SSIM index represents another step toward the replacement of the MSE with SSIM in image processing applications